Implementation of supersymmetric processes in the HERWIG event generator

JHEP04(2002)028

Received: April 12, 2002 Accepted: April 17, 2002

Implementation of supersymmetric processes in the

HERWIG event generator

Stefano Moretti

Theory Division, CERN and IPPP, University of Durham, UK E-mail: [email protected]

Kosuke Odagiri

Theory Group, KEK, Japan E-mail: [email protected]

Peter Richardson

Cavendish Laboratory and DAMTP, University of Cambridge, UK E-mail: [email protected]

Michael H. Seymour

Department of Physics & Astronomy, University of Manchester, UK E-mail: [email protected]

Bryan R. Webber

Cavendish Laboratory, University of Cambridge, UK E-mail: [email protected]

Abstract:We describe the implementation of supersymmetric processes in the HERWIG

Monte Carlo event generator. We define relevant parameter and mixing conventions and list the hard scattering matrix elements. Our implementation is based on the Minimum Super-symmetric Standard Model, with the option of R-parity violation. The sparticle spectrum is completely general. Both hadron-hadron and lepton-lepton collisions are covered. This

article supplements a separate publication in which the general features of HERWIG 6.2

are described, and updates the treatment of supersymmetry to version 6.4.

Keywords: Supersymmetry Phenomenology, Supersymmetric Standard Model, Higgs Physics.

∗_{Work supported in part by the U.K. Particle Physics and Astronomy Research Council and by the EU}

JHEP04(2002)028

Contents

1. Introduction 1

2. Parameter and mixing conventions 2

2.1 Particle content 3

2.2 Parameter conventions 4

2.3 Mixing conventions 5

2.3.1 Charginos 5

2.3.2 Neutralinos 8

2.3.3 Left-right sfermion mixing 10

2.3.4 Higgs bosons 11

3. Matrix elements 12

3.1 Notation 12

3.2 Sparton pair production 13

3.3 Slepton pair production 13

3.4 Gaugino production 14

3.5 Neutral Higgs production 16

3.6 Charged Higgs production 21

3.7 R-parity violating processes 23

3.7.1 Single gaugino production 24

3.7.2 Slepton production in association with a gauge boson 25

3.7.3 Slepton production in association with a Higgs boson 25

3.7.4 SMparticle production by sparticle exchange 26

3.8 Decay matrix elements 27

4. Conclusions 27

A. Process codes 28

1. Introduction

HERWIG1 [1] is a general-purpose event generator for high-energy processes. A major

new feature is the inclusion of supersymmetric (SUSY) processes, made available for the

first time in HERWIG 6.1 and extended in subsequent versions. This article describes

the implementation, defines the parameter and mixing conventions, and lists the hard scattering matrix elements. For the details of the input file format and for a more general

description of our implementation, as well as the description of the Standard Model (SM)

1

JHEP04(2002)028

physics implemented in the generator, the reader should consult ref. [1]. This article supplements that paper and extends it to version 6.4. The exact version number of the code described in this article is 6.400.

The key features of theHERWIG6.4 supersymmetry implementation are as follows:

• A general Minimal Supersymmetric Standard Model (MSSM) particle spectrum.

• Both hadron-hadron and lepton-lepton collisions.

• MSSM 2_→2 sparticle production subprocesses.

• 2 _→ 1, 2 _→ 2 and 2 _→ 3 Two Higgs Doublet Model (2HDM) Higgs boson

produc-tion subprocesses in hadron-hadron collisions, and 2 _→ 2 Higgs boson production

processes in lepton-lepton collisions.

• All R-parity violating 2 _→ 2 scattering subprocesses in hadron-hadron collisions

that can, but do not necessarily, have ans-channel resonance, and 2 _→2 scattering

processes in lepton-lepton collisions, not necessarily with ans-channel resonance.

• Cascade decay of heavy objects according to branching fractions taken from an input

data file, incorporating 3-body and 4-body matrix elements.

• Colour coherence between emitted partons.

• Spin correlations in 2_→2 R-parity conserving production processes and all cascade

decay processes.

• Beam polarisation in lepton-lepton collisions.

The masses, total decay widths and branching fractions, the mixings of gauginos, third generation sfermions and Higgs bosons, the bilinear, trilinear and R-parity violating

trilinear couplings, are read in from a data file. A separate program (ISAWIG) is available

(from the HERWIG web page) to convert the data from ISAJET [2] and HDECAY[3] to a

form suitable forHERWIG and modify it to include the R-parity violating sector.

In our implementation, supersymmetric particles do not radiate gluons. This assump-tion is reasonable if the decay lifetimes of the coloured sparticles (spartons) are much

shorter than theQCDconfinement scale. In particular, a stable gluino Lightest

Supersym-metric Particle (LSP) is not simulated. The simulation of a long-lived light top squark

would neglect effects due to gluon emission and hadronization.

CP-violating phases are not included for either the mixings or the couplings. R-parity violating lepton-gaugino and slepton-Higgs mixing is not considered.

2. Parameter and mixing conventions

There are several different conventions that are commonly adopted in the literature for

the MSSM lagrangian. While most authors have chosen to follow the conventions used in

JHEP04(2002)028

spin 0 spin 1/2 spin 1

down-type squarks down-type quarks

(deL,deR,esL,seR,eb1,eb2) (d, s, b)

up-type squarks up-type quarks

(u_eL,ueR,ecL,ceR,et1,et2) (u, c, t)

charged sleptons charged leptons

(_eeL,eeR,µeL,µeR,τe1,eτ2) (e−, µ−, τ−)

sneutrinos neutrinos

(_eνe,eνµ,νeτ) (νe, νµ, ντ)

gluino gluon

e

g g

neutral Higgs neutralinos photon andZ0

(h0_{, H}0_{, A}0_{) χe}0

i (i= 1−4) (γ, Z0)

charged Higgs charginos W±

H+ χ_e+_i (i= 1₋2) W+

Goldstino

e

G

Table 1:Particle content of the MSSM, expressed in terms of its mass eigenstates. The GoldstinoGe is the spin-1/2 component of the gravitino. We neglect mixing for the first two generation sfermions.

we have used in general follow those of refs. [4, 5]. In this section we present our conventions for the mixing matrices for the charginos, neutralinos and sfermions, together with the conversion between the conventions of refs. [2, 4, 5].

2.1 Particle content

TheMSSMparticle content, expressed in terms of its mass eigenstates, is given in table 1.

In defining our conventions we need to write the particle content in terms of the

interaction eigenstates. This is shown in table 2. Superscriptscrefer to charge conjugation,

which, stated in the four-component notation, is defined by:

ψc=Cψ¯T =iγ2γ0ψ¯T . (2.1)

We adopt the chirality basis. The two-component case, which we refer to when dis-cussing mixing conventions, follows trivially. Acting on chirality-left and chirality-right two-component spinors, we have:

(φL)c = iσ2φ∗L=εijφ∗jL,

(φR)c = −iσ2φ∗R=−εijφ∗jR. (2.2)

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spin 0 spin 1/2 spin 1

e

Q=µ eU_eL

DL

¶

e

D=De∗ R

e

U =Ue∗ R Q= µ_u L dL ¶

D =dc_R U =uc

R e L= µ e ν e eL ¶ e

E =_ee∗ R L= µ ν eL ¶

E =ec_R

e g g e B f W =   f W+ f

W₃0

f W−   B W =   W+

W₃0 W−

 

H1=

µ_H0 1

H₁−

¶ e

H1=

µ eH0 1

e

H₁−

¶

H2=

µ

H₂+ H0 2

¶ e

H2=

µ eH₂+

e H0 2 ¶ e G

Table 2: Field content of the MSSM, expressed in terms of its interaction eigenstates. Generation and colour indices are suppressed.

2.2 Parameter conventions

In terms of the scalar fields defined in table 2, we write our scalar superpotential as follows:

Ws=εij

h

µH₁iH₂j +f`H1iLejEe+fDH1iQejDe −fUH2iQejUe

i

. (2.3)

The above definition is consistent with refs. [5, 6]. The sign of the term proportional to the

up-type quark coupling Yukawa matrixfU is such that the Yukawa couplings are positive.

For the R-parity violating scalar superpotential, we take:

W_6Rp =εij

·

κaLeiaH2j+

1

2λabcLe

i

aLejbEec+λabc0 LeiaQejbDec

¸

+1

2λ

00

abcεc1c2c3Ue

c1

a Debc2De c3

c . (2.4)

a, b, c are the generation indices. In the last term, c1, c2, c3 are the triplet colour indices

and the totally antisymmetric tensor is defined byε123= 1. Summation over colour indices

is implicit in the other terms. We omit the bilinear term proportional to κa.

We introduce our softSUSY-breaking bilinear and trilinear terms as follows:

VI=εij

h

µBH₁iH₂j +f_`AτH1iLejEe+fDAbH1iQejDe−fUAtH2iQejUe

i

+ (h.c.). (2.5)

This is consistent with the definition in ref. [5]. We neglect A term contributions to the

first two generations. The contribution of the potential to the lagrangian is _L = ₋V in

JHEP04(2002)028

Our soft SUSY-breaking masses are defined schematically as follows:

VM =

X

φ

M_φ2φ†φ+X

ψ

Mψψψ ,¯ (2.6)

where φ are the sfermion fields and ψ are the gaugino and gluino fields in the

four-component notation. Summations over colour and weak isospin indices are implicit. We

write the gaugino soft-breaking masses as M_Be =M1 and M_fW = M2. In the notation of

refs. [4, 5],M0 _{and M are used instead ofM}

1 andM2, respectively. However, the notation

M1 and M2 has become more common.

The above conventions for the signs of the µ and the A terms are the same as those

adopted in ISAJET [2] and Haber and Kane [4]. This also agrees with the conventions

of the other major SUSY event generators, SPYTHIA [7] and SUSYGEN [8]. While these

conventions are the same internally,ISAJETuses a different notation which corresponds to

the following:

2m1 = −µ ,

µ1 = −M1,

µ2 = −M2, (2.7)

wherem1, µ1 and µ2 are in the notation of ISAJET.

2.3 Mixing conventions

While internallyHERWIGuses the conventions of refs. [4, 5, 6], the mass and decay spectra

are fed in from a data file. In particular, we have provided a separate code (ISAWIG) for

the conversion of data from ISAJET[2]. This way, the masses of the sparticles and their

R-parity conserving decay rates are calculated using ISAJET and the R-parity violating

section is added byISAWIG.ISAJETalso calculates the mixing matrices for the electroweak

gauginos and scalar fermions. ISAJETuses the conventions of refs. [2, 10] and it is essential

that the conversion between the two formalisms is done correctly.

We first consider the conversion between the two formalisms for the gauginos and then for the left-right sfermion mixing. We compare the lagrangian of ref. [2] against that of refs. [4, 5] to obtain the conversions between them. We adopt the conventions of refs. [4, 5] for the chargino and neutralino mixing matrices.

2.3.1 Charginos

In the notation of ref. [5], we define the following doublets of two-component spinors for the charginos in their interaction eigenstates:

ψ+ = ³₋iλ+, ψ_H+₂´,

ψ− = ³₋iλ−, ψ_H−₁´, (2.8)

where λ± _{are the charged Winos, ψ}−

H1 is the charged Higgsino associated with the Higgs

JHEP04(2002)028

the Higgs doublet that gives mass to the up-type quarks. The mass term in the lagrangian from ref. [5, eq. (A.2)] is:

Lchargino=₋1 2

¡

ψ+ ψ−¢

µ _{0 X}T

X 0

¶ µ_ψ+

ψ−

¶

+ (h.c.), (2.9)

where the mass matrix is given by:

X =

µ

M2 MW√2 sinβ

MW √

2 cosβ µ

¶

. (2.10)

In ref. [5] this is diagonalized in the two-component notation by defining the mass eigen-states:

χ+_i = Vijψ_j+,

χ−_i = Uijψ−j , (2.11)

whereU andV are unitary matrices chosen such that

U∗XV−1=MD. (2.12)

MD is the diagonal chargino mass matrix. The four-component mass eigenstates, the

charginos, are defined in terms of the two-component fields as:

e

χ+₁ =

µ

χ+₁

(χ−₁)c

¶

, χ_e+₂ =

µ

χ+₂

(χ−₂)c

¶

. (2.13)

We need to express the chargino mass terms in the lagrangian in a four-component notation so that we can compare this lagrangian with the conventions of ref. [2]. We define the four-component spinors as in ref. [5]:

f

W+=

µ

−iλ+

(₋iλ−₎c

¶

, He+=

µ

ψ_H+₂

(ψ−_H1)c

¶

. (2.14)

Using this notation, we can express the lagrangian in the four-component notation:

Lchargino=₋³fW+ _He+´ £_XP

L+XTPR¤

µ fW+

e

H+

¶

. (2.15)

The chirality projection operators are defined asP_L/R = (1–/+γ5)/2. Equation (2.15)

has a similar form to the chargino mass term2 _{in ref. [2, eq. (2.1)]:}

Lchargino=₋¡λ¯ χ¯¢ £MchargePL+MchargeT PR¤

µ

λ χ

¶

, (2.16)

where

Mcharge =

µ

µ2 −gv0 −gv 2m1

¶

=₋

µ

M2 √2MWcosβ √

2MWsinβ µ

¶

, (2.17)

JHEP04(2002)028

wherev andv0 are the vacuum expectation values for the Higgs fields that give mass to the

up- and down-type quarks, respectively. We now come to the problem of comparing the notation of ref. [5] against that of ref. [2]. The Wino and Higgsino fields of ref. [2] are the charge conjugates of those used in ref. [5], so that the following transformation is required:

f

W+ = λc,

e

H+ = χc. (2.18)

This gives the chargino mass matrix in the form:

Lchargino =³fW+_,He+´ h_M0

chargePL+M0TchargePR

i µ fW+

e

H+

¶

, (2.19)

where

M_charge0 =

µ

−µ2 gv

gv0 _−2m 1 ¶ = µ M2 √

2MWsinβ

√

2MW cosβ µ

¶

. (2.20)

This agrees with eq. (2.15) apart from the overall sign. We need to express the fields in

terms of the ISAJET mixing matrices. The ISAJET mixing matrices are given in ref. [2,

eqs. (2.10) and (2.11)] as:

µ fW+

f W₋ ¶ L = µ

θxcosγL −θxsinγL

sinγL cosγL

¶ µ λ χ ¶ L , µ

(₋1)θ+_Wf

+

(₋1)θ−f_W

−

¶

R

=

µ

θycosγR −θysinγR

sinγR cosγR

¶ µ λ χ ¶ R . (2.21)

The mixing angles γL and γR, and the sign functions θx, θy, θ+, and θ− are defined in

ref. [2].

We can transform these equations into the notation of ref. [4] with the identification:

e

χ+₁ =Wf₋c,

e

χ+₂ =Wf₊c. (2.22)

This gives:

PL

µ fW+

e

H+

¶

=PL

µ

−sinγR −θycosγR −cosγR θysinγR

¶ µ

(₋1)θ−_χe+

1

(₋1)θ+_χe+

2

¶

,

PR

µ fW+

e

H+

¶

=PR

µ

−sinγL −θxcosγL −cosγL θxsinγL

¶ µ e

χ+₁

e

χ+₂

¶

. (2.23)

These mixing matrices are now in a form that allows us to compare them with the notation of ref. [5, eq. (A.13)], :

PL

µ fW+

e

H+

¶

= PL

µ

V∗ 11 V21∗

V∗ 12 V22∗

¶ µ e

χ+₁

e

χ+₂

¶

,

PR

µ fW+

e

H+

¶

= PR

µ

U11 U21

U12 U22

¶ µ e

χ+₁

e

χ+₂

¶

JHEP04(2002)028

By comparing eqs. (2.23) and (2.24) we obtain the mixing matrices in the notation of ref. [5]:

U =

µ

−sinγL −cosγL −θxcosγL θxsinγL

¶

,

V =

µ

−sinγR(−1)θ− −cosγR(−1)θ+ −θycosγR(−1)θ− θysinγR(−1)θ+

¶

. (2.25)

It should be noted that we adopt the opposite sign convention for the chargino masses due to the sign differences in the two lagrangians.

2.3.2 Neutralinos

We define a quadruplet of two-component fermion fields for the neutralinos in the interac-tion eigenstate as:

ψ0 =¡₋iλ0,₋iλ3, ψ0_H1, ψ0_H2¢, (2.26)

where λ0 _{is the Bino, λ}3 _{is the neutral Wino, ψ}0

H1 is the Higgsino corresponding to the

neutral component of the Higgs doublet giving mass to the down-type quarks and ψ0_H2 is

the Higgsino corresponding to the neutral component of the Higgs doublet giving mass to the up-type quarks. The lagrangian for the neutralino masses from ref. [5, eq. (A.18)] is:

Lneutralino=₋1 2

¡

ψ0¢T Y ψ0+ (h.c.), (2.27)

where

Y = (2.28)

=

   

M1 0 −MZsinθWcosβ MZsinθWsinβ

0 M2 MZcosθWcosβ −MZcosθWsinβ

−MZsinθWcosβ MZcosθWcosβ 0 −µ

MZsinθW sinβ −MZcosθWsinβ −µ 0

   .

In ref. [5], the lagrangian is diagonalized in this two-component notation to give the mass eigenstates. The diagonalization is performed by defining two-component fields:

χ0_i =Nijψj0, i, j = 1, . . . ,4, (2.29)

whereN is a unitary matrix satisfyingN∗_{Y N}−1 _=N

D, andND is the diagonal neutralino

mass matrix. The four-component mass eigenstates, the neutralinos, can be defined in terms of the two-component fields:

e

χ0_i =

µ

χ0_i

(χ0

i)c

¶

. (2.30)

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in the four-component notation by defining four-component Majorana fields:

e

B =

µ

−iλ0

(₋iλ0₎c

¶

, Wf3 =

µ

−iλ3

(₋iλ3)c

¶

,

e

H1 =

µ

ψ_H0₁

(ψ0

H1)

c

¶

, He2=

µ

ψ_H0₂

(ψ0

H2)

c

¶

. (2.31)

As defined in table 2, Be is the Bino field, fW3 is the neutral Wino field, He1 is the field for

the Higgsino associated with the neutral component of the Higgs doublet that gives mass

to the down-type quarks and He2 is the field for the Higgsino associated with the neutral

component of the Higgs doublet that gives mass to the up-type quarks. This gives the lagrangian in the four-component notation:

Lneutralino =₋1 2

³ e

B Wf3 He1,He2

´

[Y PL+Y PR]

    e B f W3 e H1 e H2   

. (2.32)

We can now compare this with the relevant lagrangian, in the four-component notation,

given in ref. [2]. This lagrangian3 is from ref. [2, eq. (2.2)]:

Lneutralino =₋1 2

³

¯

h0,¯h00,λ¯3,λ¯0

´

[MneutralPL+MneutralPR]

   

h0 h00

λ3

λ0

  

. (2.33)

h0 is the Higgsino partner of the neutral component of the Higgs doublet that gives mass

to the up-type quarks,h00_{is the Higgsino partner of the Higgs boson that gives mass to the}

down-type quarks,λ3 is the neutral Wino andλ0 is the Bino. The mass matrix is given by:

Mneutral =

   

0 ₋2m1 −gv/√2 g0v/√2

−2m1 0 gv0/ √

2 ₋g0v0/√2

−gv/√2 gv0_/√_{2 µ}

2 0

g0_v/√_{2 −g}0_v0_/√_{2 0 µ} 1

  

. (2.34)

It is easier to compare this with eq. (2.32) after reordering the entries and reexpressing

it in terms ofMZ, β and θW. This gives:

Lneutralino = 1 2

³

¯

λ0,¯λ3,h¯

0₀

,¯h0´ £M_neutral0 PL+Mneutral0 PR¤

    λ0 λ3

h00

h0

  

, (2.35)

where:

M_neutral0 = (2.36)

=

   

M1 0 MZsinθWcosβ −MZsinθWsinβ

0 M2 −MZcosθW cosβ MZcosθW sinβ

MZsinθWcosβ −MZcosθW cosβ 0 −µ −MZsinθWsinβ MZcosθWsinβ −µ 0

   .

JHEP04(2002)028

This equation then agrees with eq. (2.32) up to an overall sign, provided we make the following identification:

e

B =λ0, Wf3 =λ3,

e

H1 =−h00, He2 =−h0. (2.37)

The convention fromISAJETfor the mixing, taken from ref. [2, eq. (2.12)], is

   

(₋iγ5)θ1Ze1

(₋iγ5)θ2Ze2

(₋iγ5)θ3Ze3

(₋iγ5)θ4Ze4

   =

   

v(1)₁ v₂(1) v₃(1) v₄(1) v(2)₁ v₂(2) v₃(2) v₄(2) v(3)₁ v₂(3) v₃(3) v₄(3) v(4)₁ v₂(4) v₃(4) v₄(4)

   

   

h0

h00 λ3

λ0

  

, (2.38)

where Zei are the mass eigenstates obtained by diagonalizing the mass matrix given in

eq. (2.34), v_j(i) are the elements of the mixing matrix, andθi is zero (one) if the mass ofZei

is positive (negative).

After reordering, we get:

   

(₋iγ5)θ1Ze1

(₋iγ5)θ2Ze2

(₋iγ5)θ3Ze3

(₋iγ5)θ4Ze4

   =

   

v(1)₄ v(1)_{3 −}v₂(1) ₋v(1)₁ v(2)₄ v(2)_{3 −}v₂(2) ₋v(2)₁ v(3)₄ v(3)_{3 −}v₂(3) ₋v(3)₁ v(4)₄ v(4)_{3 −}v₂(4) ₋v(4)₁

   

   

e

B

f

W

e

H1

e

H2

  

. (2.39)

We can therefore obtain the mixing matrix in the notation of ref. [5] by making the iden-tification:

Ni1 = v(i)4 , Ni2 =v3(i),

Ni3 = −v2(i), Ni4 =−v(i)1 . (2.40)

Again, we need to adopt the opposite sign convention for the neutralino masses.

2.3.3 Left-right sfermion mixing

In addition to the mixing of the neutralinos and charginos, we need to consider the left-right mixing of the sfermions. In general, as the off-diagonal terms in the mass matrices are proportional to the fermion mass, these effects are only important for the third generation sfermions, i.e. stop, sbottom and stau. We describe our convention for the top squarks. The others are treated analogously.

The following mass matrix for the top squarks uses the conventions4 _{of ref. [5] and is}

taken from eq. (4.17) therein.

M_et2= (2.41)

=

µ_M2

e

Q+M

2

Zcos 2β

¡₁

2 −23sin2θW

¢

+m2

t mt(At−µcotβ)

mt(At−µcotβ) M_U2_e+2₃m2Zcos 2βsin2θW +m2t

¶

,

whereM_Qe andM_Ue are softSUSY-breaking masses for the left and right top squarks,

respec-tively. At is the trilinear softSUSY-breaking term for the interaction of the left and right

JHEP04(2002)028

stop squarks with the Higgs boson. This compares with the ISAJETmatrix from ref. [10]:

M_et2= (2.42)

=

µ_M2

etL+M

2

Zcos 2β

¡₁

2 −23sin2θW

¢

+m2_{t −}mt(At−µcotβ)

−mt(At−µcotβ) M_et2

R+

2

3m2Zcos 2βsin2θW +m2t

¶

,

where M2

etL = M

2

e

Q and M

2

e

tR = M

2

e

U. There is a difference in the sign of the off-diagonal

terms. This means that, as the sign conventions for theµand Aterms are the same, there

is a difference in the relative phases of the two fields in the different conventions. Hence

we should apply the following change to theISAJEToutput:

θt −→ −θt, (2.43)

i.e. change the sign of the stop mixing angle. The same argument also applies to the sbottom and stau mixing angles.

We adopt the following convention for the sfermion mixing matrices:

µ e

qiL

e

qiR

¶

=

µ

cosθi

q sinθqi −sinθi_q sinθ_qi

¶ µ e

qi1

e

qi2

¶

=

µ

Qi

L1 QiL2

Qi_R1 Qi_R2

¶ µ e

qi1

e

qi2

¶

, (2.44)

whereq_eiL and eqiR are the left and right squark fields, for theith quark, whereiruns over

quark flavours d, u, s, c, b, t. _eqi1 and eqi2 are the squark fields for the mass eigenstates, for

the quarki, and θi_q is the mixing angle obtained by diagonalizing the mass matrix.

Similarly, we denote the slepton mixing as above with the matrix Li

αβ where i runs

over the charged and neutral lepton flavour. βis the mass eigenstate andαis the left/right

eigenstate. As we do not include the right-handed neutrino we omit the left-right mixing for the sneutrinos. Note that our convention for the mixing matrix is opposite to that for

the gauginos, in the sense that here the mixing matricesQi act on mass eigenstates to give

the interaction eigenstates.

2.3.4 Higgs bosons

The Higgs doublets are diagonalised as follows. As in tables 1 and 2,Hi denote the SU(2)

doublet interaction eigenstates which form the physical eigenstates h0_{, H}0_{, A}0_{, H}± _after

the electroweak symmetry breaking. In the charged sector:

H₂+=H+cosβ+G+sinβ , H₁−=H−sinβ₋G−cosβ , (2.45)

where G± _{are the Goldstone modes. The neutral Higgs sector is parametrised in the}

interaction eigenstate as follows:

H₁0 = vcosβ+_√η1+iζ1

2 ,

H₂0 = vsinβ+√η2+iζ2

2 . (2.46)

v is the vacuum expectation value. The diagonalisation is achieved by:

ζ2 =A0cosβ+G0sinβ , ζ1=A0sinβ−G0cosβ ,

η2 =H0sinα+h0cosα , η1 =H0cosα−h0sinα . (2.47)

JHEP04(2002)028

3. Matrix elements

3.1 Notation

For 2 _→ 2 hard scattering subprocesses of the form 12 _→ 34, let us define the usual

Mandelstam variables in terms of the partonic momenta as ˆs= (p1+p2)2, ˆt= (p1−p3)2

and ˆu = (p1 −p4)2. For simplicity, the “hat” will be omitted in the following. As initial

state partons are taken to have zero kinematic mass, it follows that s+t+u=m2₃+m2₄,

s+t3+u4 = 0, and ut−m23m24 = sp2T where t3 =t−m23, u4 = u−m24, and pT is the

outgoing transverse momentum.

The electromagnetic coupling ise, such thatαEM =e2/4π. The strong coupling isgs,

such thatαs=g2s/4π. TheQCDcolour factors areNC = 3 andCF = (NC2−1)/2NC = 4/3.

Colour flow directions are not stated explicitly as they are trivial in most of the pro-cesses whose matrix elements are given below. For a general treatment, see ref. [13]. Matrix elements given here are summed over the final state colour and spin, and averaged over the initial state colour and spin. As usual, this is indicated by a long overline. Statistical factors of two for identical neutralinos in the final state are written explicitly.

Let us now consider the widths Γ(q2) appearing in the propagators of unstable massive

particles with massM(q2). In order to simplify the notation, let us define a function Γ(q2),

which is defined for positive q2 by:

p

q2_Γ(q2_)≡M(q2_)¯Γ(q2_{). (3.1)}

For negative q2_{, the width is taken to be zero. The propagator is}

f Qf I_f3

e ₋1 ₋1/2

ν 0 +1/2

d ₋1/3 ₋1/2

u +2/3 +1/2

Table 3: The U(1)EM chargeQf and the weak isospin I3

f for the Stan-dard Model fermions.

then given in general, again for positive q2_{, by:}

1

q2_−M2_(q2_{) +i}p_q2_Γ(q2₎ ≡

1

q2_−M2_(q2_{) +iM(q}2_)Γ(q2₎. (3.2)

In theHERWIGimplementation, we adopt the naive running width

approximation and neglect the running of mass, so that: ¯

Γ(q2) = q

2_Γ(M2₎

M2 . (3.3)

The following coupling notations are used in processes involving electroweak interac-tions:

Qf =U(1)EMcharge ; (3.4)

I_f3 = weak isospin ; (3.5)

Lf =

(I_f3₋Qfsin2θW)

cosθW sinθW

; (3.6)

Rf = −

Qfsin2θW

cosθWsinθW

. (3.7)

JHEP04(2002)028

sfermion coupling coefficients as:

e

Lf a = (Na1cosθW +Na2sinθW)Qf + (−Na1sinθW +Na2cosθW)Lf;

e

Rf a = (Na1cosθW +Na2sinθW)Qf + (−Na1sinθW +Na2cosθW)Rf. (3.8)

Following ref. [4], we have neglected the contribution of the Higgsino component, which is proportional to the Yukawa coupling. This affects the cross sections for sfermion-Higgsino

associated production, namely the Higgsino component of the processesgb_→ etχ_e±_,e_bχe0_.

3.2 Sparton pair production

The matrix elements and the colour decompositions for 2_→2 parton-to-sparton scattering

are given in ref. [13]. The term “sparton” refers to coloured superparticles, namely the squarks and the gluino. Neglecting the squark left-right mixings, the lepton-annihilation

reaction `−<sub>`</sub>+_→qeqe∗ _{is given by:}

|M|2

(L,R) = e4NCsp2T × ×   ¯ ¯ ¯ ¯ ¯

Q`Qq

s +

L`(Lq, Rq)

s₋M_Z2 +iMZΓZ

¯ ¯ ¯ ¯ ¯ 2 + ¯ ¯ ¯ ¯ ¯

Q`Qq

s +

R`(Lq, Rq)

s₋M_Z2 +iMZΓZ

¯ ¯ ¯ ¯ ¯ 2

. (3.9)

The generalization to the case with left-right mixings is given by:

(Lq, Rq)−→LqQLiQLj+RqQRiQRj. (3.10)

The indicesi, j label the squarks that are produced.

3.3 Slepton pair production

The matrix elements are similar to eqs. (3.9) and (3.10). For the quark annihilation reaction

qq¯_→ `e`e∗ _{one should interchange the indices (q ↔ `), and change the colour factor from}

NC to 1/NC. In addition, for the charged current reaction qq¯0 →e`Leν∗ we have

|M|2_(qq¯0 _→e` Leν∗) =

e4sp2_T

¡

2 sin2θW¢2NC

¯ ¯ ¯ ¯ ¯ 1

s₋M2

W +iMWΓW

¯ ¯ ¯ ¯ ¯ 2 . (3.11)

For the lepton annihilation reaction the colour factor is 1, and there is an extrat-channel

gaugino contribution to the production of slepton pairs of the same generation as the colliding lepton pair, namely,

|M|2_(e−_e+

→eeLee∗L,eeRee∗R) = e4sp2T × (3.12)

×   ¯ ¯ ¯ ¯ ¯ 1 s+

Le(Le, Re)

s₋M2

Z+iMZΓZ

+ 4 X a=1 Ã e L2 ea

t₋M2

e

χ0 a

,0!¯¯¯_¯

¯ 2 + + ¯ ¯ ¯ ¯ ¯ 1 s+

Re(Le, Re)

s₋M2

Z+iMZΓZ

+

4

X

a=1

Ã

0, Re

2 ea

t₋M2

e χ0 a !¯¯_¯ ¯ ¯ 2 ,

|M|2_(e−_e+

→eeLee∗R,eeRee∗L) = e4

à ₄ X

a=1

M_χe0 a

√

s t₋M2

e

χ0 a

e

LeaReea

!2

JHEP04(2002)028

The neutralino masses appearing above are the signed masses. The production of sneutrinos of the same generation as the colliding lepton pair is given by:

|M|2_(e−_e+

→eνeLνeeL∗ ) = e4sp2T

  ¯ ¯ ¯ ¯ ¯

LeLν

s₋M2

Z+iMZΓZ

+

2

X

a=1

(V_a1χe±)2/2 sin2θW

t₋M2

e

χ±a

¯ ¯ ¯ ¯ ¯

2

+

+

¯ ¯ ¯ ¯ ¯

ReLν

s₋M2

Z+iMZΓZ

¯ ¯ ¯ ¯ ¯

2

. (3.14)

3.4 Gaugino production

We consider the following seven processes in hadronic collisions:

qq¯_−→ χ_e+_aχ_e−_b , (3.15)

qq¯_−→ χ_e0_iχ_e0_j, (3.16)

qq¯0 _−→ χ_e±_aχ_e0_i, (3.17)

qq¯_−→ χ_e0_ieg , (3.18)

gq _−→ χ_e0_ieq and c.c., (3.19)

qq¯0 _−→ χ_e±_aeg , (3.20)

gq _−→ χ_e±_aq_e0 and c.c. (3.21)

c.c. refers to charge conjugation. The indices take the valuesa, b= 1,2 andi, j = 1, . . . ,4.

Only processes (3.15) and (3.16) are relevant to leptonic collisions, where the colour factors and electroweak coupling coefficient need trivial modifications. Processes (3.15), (3.16), (3.17), (3.18) and (3.20) are expressed conveniently following the notation of ref. [14]:

A= X

α,β=L,R

Cαβ(¯v2γµPαu1)(¯u3γµPβv4) =

X

α,β=L,R

CαβQµαXβµ, (3.22)

where, as before,PL = (1−γ5)/2 andPR= (1 +γ5)/2. The spin-averaged amplitudes are

given as:

|A_|2 _{= [|C}

LL|2+|CRR|2]u3u4+ [|CLR|2+|CRL|2]t3t4+

+ 2 Re[C_LL∗ CLR+CRR∗ CRL](m3m4s). (3.23)

The masses appearing above as (m3m4s) are the signed masses.

The matrix elements squared for processes (3.19) and (3.21) have the following form,

whereα labels the chirality of the final state squark:

|M|2

α=

(gsgα)2

2NC

·

−t_s3 − 2(m 2

4−m23)t3

su4

µ

1 + m

2 3

t3

+m

2 4

u4

¶¸

. (3.24)

When left-right squark mixing is considered, the above is modified as:

|M|2

JHEP04(2002)028

Process (3.15),qq¯_→χ_e+_aχ_e−_b, is given by eq. (3.23) with the prefactor (e4/NC) and the

following coefficientsCαβ:

Cαβ =−δab

Qq

s −

1

sin 2θW ·

(δαLLq+δαRRq)(−2O_ab0β)

s₋M_Z2 +iMZΓZ − − δαL

2 sin2θW

"

(1₋δαβ)δquUa1Ub1

t₋M2

e

dL

−δαβδqdVa1Vb1

u₋M2

e

uL

#

. (3.26)

Here, following the notation of ref. [4], we have defined:

−2O0L

ab = −Va2Vb2+ 2δabcos2θW;

−2O0R_ab = ₋Ua2Ub2+ 2δabcos2θW. (3.27)

Process (3.16), qq¯ _→ χ_e0

iχe0j, is given by eq. (3.22) with the prefactors (e4/NC) and

1/(1 +δij), and the following coefficients Cαβ:

CLL =

Lq

sin 2θW ·

Ni4Nj4−Ni3Nj3

s₋M_Z2 +iMZΓZ

+ LeqiLeqj

u₋M_eq2

L

;

CLR =

Lq

sin 2θW ·

Ni3Nj3−Ni4Nj4

s₋M2

Z+iMZΓZ

− _tLeqiLeqj −M2

e

qL

;

CRL =

Rq

sin 2θW ·

Ni4Nj4−Ni3Nj3

s₋M2

Z+iMZΓZ −

e

RqiReqj

t₋M_qe2

R

;

CRR =

Rq

sin 2θW ·

Ni3Nj3−Ni4Nj4

s₋M_Z2 +iMZΓZ

+ ReqiReqj

u₋M2

e

qR

. (3.28)

Process (3.17),ud¯_→χ_e+

aχe0i, is given by eq. (3.22). The prefactors are given by (e4/NC)

and (_|VCKM|2/2 sin2θW) whereVCKMrefers to the relevant term of the

Cabbibo-Kobayashi-Maskawa (CKM) matrix, and the coefficients Cαβ are:

CLL =

1

sinθW ·

Ni2Va1−Ni4Va2/√2

s₋M2

W +iMWΓW

+ Va1Leui

u₋M_ue2

L

;

CLR =

1

sinθW ·

Ni2Ua1+Ni3Ua2/√2

s₋M2

W +iMWΓW

− Ua1Ledi

t₋M2

e

dL

;

CRL = 0 ;

CRR = 0. (3.29)

Process (3.18), qq¯_→ χ_e0_ieg, is given by eq. (3.22) with the prefactor (e2g_s2CF/NC) and

the following coefficients Cαβ:

CLL =

e

Lqi

u₋M2

e

qL

; CLR =−

e

Lqi

t₋M2

e

qL

;

CRL =

e

Rqi

t₋M_eq2

R

; CRR=−

e

Rqi

u₋M_qe2

R

JHEP04(2002)028

Process (3.19),gq_→χ_e0_iq_e, is given by eq. (3.24), with gα =e(δαLLeqi−δαRReqi).

Process (3.20),ud¯_→χ_e+_aeg, is given by eq. (3.22). The prefactors are (e2g_s2CF/NC) and

(_|V_CKM|2_{/2 sin}2_θ

W), and the coefficients Cαβ are:

CLL =

Va1

u₋M_ue2

L

; CLR=−

Ua1

t₋M2_e

dL

;

CRL = 0 ; CRR= 0. (3.31)

Process (3.21),gq_→χ_e±

aqeL0 , is given by eq. (3.24), withgL=e(δquUa1+δqdVa1).

Processes (3.17), (3.20) and (3.21) involve flavour-changing CKM matrix contributions. We assume that the CKM matrix is universal and applicable also to sfermion interactions. Process (3.15) involves two flavour-changing vertex contributions and, in principle, we should sum over the internal squark flavours and external quark pairs. However, we believe that to a good approximation this process can be treated correctly by substituting

an identity matrix for the CKM matrix and so this is done in HERWIG.

3.5 Neutral Higgs production

For hadronic collisions, the following five classes of production subprocesses are imple-mented:

(a) gluon-gluon fusion via loops of heavy quarks Q=b and tand squarks Qe=eband et,

(b) W±_W∓_{, Z}0_Z0 _fusion,

(c) associated production withW±_{, Z}0 _bosons,

(d) associated production with pairs of heavy quarksQQ¯,

(e) associated production with pairs of heavy squarksQeQe∗_.

The scalar Higgs bosonsh0 _{and H}0 _{can be produced by all five subprocesses (a)–(e),}

whereas the pseudoscalarA0 is only produced via the first and the last two, as theA0V V

vertex, withV =W±_{, Z}0_{, is absent at tree level. Higgs bosons are also produced in decays}

of heavier sparticles.

For leptonic collisions, the following are implemented:

(f) W±_W∓_{, Z}0_Z0 _fusion,

(g) associated production withZ0_,

(h) h0A0 and H0A0 production.

Of these, (f) and (g) are expressed by formulae that are obtained by changing theZ0boson

couplings in the corresponding hadronic subprocesses. In addition, for (g), the colour factor

JHEP04(2002)028

Higgs boson gu,ν gd,` gW±_,Z0

SM φ 1 1 1

MSSM h0 _{cosα/sinβ −sinα/cosβ sin(β−α)}

H0 sinα/sinβ cosα/cosβ cos(β₋α)

A0 _{1/tanβ tanβ 0}

Table 4: Higgs coupling coefficients in the MSSM to isospin +1/2 and−1/2 fermions and to the gauge bosonsW±_{, Z}0_{, as first appearing in eqs. (3.32) and (3.33).}

In the following, let us denote the neutral Higgs bosons collectively by Φ. We define

the following notation for invariant mass squared. M2

[ab] is the invariant mass squared

of the particle pair [ab], M_[a2

ibi] = M

2

[12] = (pa +pb)2 = s, M[a2fbf] = (pa +pb)

2 _and

M2

[aibf]= (pa−pb)

2_{, where the subscripts iandf indicate particles in the initial and final}

states, respectively. M_[ab]2 is negative only in the last of these three cases.

For the gluon-fusion subprocess class (a), the matrix elements are cast in the form seen in ref. [15]:

|M|2_{(gg →h}0_{, H}0_{) =} αEMα 2

SM

4 h,H

72πsin2θW(NC2 −1)MW2

×

×

¯ ¯ ¯ ¯ ¯ ¯

X

f

g_fh,HAh,H_f (τf) +

X

e

f

gh,H_e

f A

h,H

e

f (τfe)

¯ ¯ ¯ ¯ ¯ ¯

2

, (3.32)

|M|2_(gg→A0_{) =} αEMα2SMA4

72πsin2θW(N_C2 −1)M_W2

¯ ¯ ¯ ¯ ¯ ¯

X

f

gA_fAA_f(τf)

¯ ¯ ¯ ¯ ¯ ¯

2

, (3.33)

for scalar and pseudoscalar production, respectively. The couplings gf and g_f˜ are those

appearing in tables 4 and 5, respectively. We have defined τ = 4m2_/M2_{, where M is the}

mass of the Higgs boson concerned and mis the mass of the particle inside the loop. The

functionsA are given by:

Ah,H_Q (τ) = 3

2τ[1 + (1−τ)f(τ)],

Ah,H_e

Q (τ) = −

3

4[1−τ f(τ)], (3.34)

for the scalars and

AA_Q(τ) =τ f(τ), (3.35)

for the pseudoscalar, with the function f(τ) given by

f(τ) =

  

arcsin2 1√

τ, τ ≥1 ,

−14

h

log1+√1−τ

1−√1−τ −iπ

i2

JHEP04(2002)028

Higgs boson g_fe

i

SM φ 0

MSSM h0 m

−2

e

fi

h

m2_fgh_f +M_Z2cosθWsinθW(LfQ2Li−RfQ2Ri) sin(α+β)

+ (µmf(δf usin_sinα_β −δf d_coscosα_β) +Afmfgh_f)QLiQRi

i

H0 m

−2

e

fi

h

m2_fgH_f +M_Z2cosθWsinθW(RfQ2Ri−LfQ2Li) cos(α+β)

+(µmf(−δf ucossinβα −δf dcossinαβ) +AfmfgfH)QLiQRi

i

A0 0

Table 5: MSSM Higgs coupling coefficients to sfermions ˜f = ˜q,`˜, as first appearing in eq. (3.32). For the leptonic case, the mixing matrices Qij should be replaced by the corresponding slepton mixing matricesLij.

For process (b), the matrix element for theW±_W∓ _{fusion process is given by:}

|M|2_(q

1q02→q300q4000Φ5) = (gΦW±)2

µ

e2

2 sin2θW

¶3

(2M_W2 )_×

×|VCKM[qq 00_]|2_|V

CKM[q0q000]|2M_[12]2 M_[34]2

(M2

[13]−MW2 )(M[24]2 −MW2 )

, (3.37)

whereas forZ0Z0 fusion we have:

|M|2_(q

1q02→ q3q40Φ5) = (gZΦ0)2

µ

e2

4 sin2θW cos2θW

¶3

(4M_Z2)_× (3.38)

×(L 2

qL2q0 +R2_qR2_q0)M_[12]2 M_[34]2 + (L2_qR2_q0 +R2_qL2_q0)M_[14]2 M_[23]2

(M2

[13]−MZ2)(M[24]2 −MZ2)

.

The coupling coefficients g_VΦ are as given in table 4. The formulae are identical to those

for the case of leptonic collisions, with the replacement of the CKM matrix elements by

unity for theW±_W∓_{fusion subprocess and the substitution of the couplingsL, Rwith the}

appropriate leptonic couplings Le, Re for theZ0Z0 fusion subprocess.

For process (c), the matrix elements are given by:

|M|2_(u

1d¯2 →W3+Φ4) = (g_WΦ±)2

1

NC

µ

e2

2 sin2θW

¶2µ

1 2

¶

|VCKM[ud]|2×

× sp

2

T + 2MW2 s

(s₋M_W2 )2_+M2

WΓ

2 W

, (3.39)

|M|2_(q

1q¯2 →Z3Φ4) = (g_ZΦ0)2

1

NC

µ

e2

4 sin2θWcos2θW

¶2

(L2_q+R_q2)_×

× sp

2

T + 2MZ2s

(s₋M2

Z)2+MZ2Γ 2 Z

JHEP04(2002)028

The coupling coefficients gΦ_V are again as given in table 4. The above formula, for the Z0

mediated subprocess, is immediately applicable to the leptonic case with the substitution of

the colour factor 1 to replace 1/NC. The couplingsLq, Rqare replaced byLe, Rein this case.

For process (d), the matrix element for theqq¯initiated subprocess is given by:

|M|2_(q

1q¯2 →Q3Q¯4Φ5) =

e2g_s4(gΦ_Q)2

2 sin2θWM_W2 s

µ

CF

NC

¶

G × (3.41)

×

·

1

2_G5 −

M2s_G

2 +M

2_(G3p2

T3+G4p2T4)− G5(s+M2)p2T5

¸

.

Here M2 = M_Φ2 for A0 and M = M_Φ2 ₋4m2_Q for h0, H0. The pT’s are the transverse

momenta of the final state particles. These can be expressed asp2

Ti = 4p1·pi p2·pi/s−m

2 i

(where i = 3,4,5) as the incoming particles are considered massless. The propagator

factors are defined by:

G3 = 1

(M2

[35]−m2Q)

,

G4 = 1

(M2

[45]−m2Q)

,

G =_G3+_G4,

G5 = G3G4

G . (3.42)

For the case Q =b, the gg initiated subprocess is evaluated with 2 _→ 1, 2 _→ 2 and

2 _→ 3 matrix elements. In order to avoid double counting, it is recommended that the

user chooses only one of these, rather than sum over them, according to their need. For

the 2_→1 subprocess the matrix element is given by:

|M|2_(b

1¯b2 →Φ5) =

e2_M2

Hm2b(gbΦ)2

8 sin2θWMW2 NC

. (3.43)

The 2_→2 matrix element is given by:

|M|2_(g

1b2 →b3Φ4) =

µ

g2 s

2NC

¶ µ

e2

2 sinθ2

W

¶ µ

m2_b(g_bΦ)2

2M2

W

¶

×

×

µ

−u2₄ st3

¶ ·

1 + 2m

2 4−m23

u4

µ

1 + m

2 3

t3

+m

2 4

u4

¶¸

. (3.44)

The inconsistency in treating the initial state bottom quark as being massless and the final state bottom quark as being massive is unavoidable. The subprocess implementation being

intended for highpT singleb-jet events only, the ambiguity in the treatment of the bottom

quark mass in the hard scattering matrix element is negligible.

We evaluated the 2_→ 3 matrix element for gg _→ QQ¯Φ using helicity amplitudes as

JHEP04(2002)028

For process (e), the matrix element for theqq¯initated subprocess is given by:

|M|2_(q

1q¯2 →Qe3Qe∗4Φ5) =

 g

4 Se2g_QΦ_e

2

M2_e

Q

sin2θWM_W2 s

 

µ

CF

NC

¶

×

×£|G3|2p2_T3+_|G4|2p2_T4−2 Re (_G∗_3G4)pT3·pT4

¤

. (3.45)

The propagator factors are given by:

G3 = ¡ 1

(p4+p5)2−m2₃+im3Γ3¢

,

G4 = ¡ 1

(p3+p5)2−m24+im4Γ4¢

. (3.46)

The dot product of two transverse momenta is two dimensional, i.e.pT3·pT4 =px3px4+

py₃py₄.

The gg initiated subprocess is again calculated using helicity amplitudes in an

opti-mized basis. This time the two orthogonal polarisation vectors for the initial state gluons

are chosen to be in thex and y directions, yielding a relatively compact expression:

|M|2_(g

1g2 →Qe3Qe∗4Φ5) =

  g

4 Se2gQΦe

2

M2_e

QNC

4 sin2θWMW2 (NC2 −1)

 _×

× X

i=x,y

X

j=x,y

·

|At(i, j)_|2+_|Au(i, j)_|2₋ 1

N2 C

|At(i, j) +

+_Au(i, j)_|2

¸

. (3.47)

The colour flow components are distributed using the method of ref. [13], namely, the

(1/N_C2) term is distributed amongst the t-channel and u-channel colour flow components

by their ratio. The two amplitude functions_At and_Au are given by:

At(i, j) = 2¡pi3pj₄G24G13+pi3pj₃G13G45+pi4pj₄G24G35¢−

−√δij

s(pz3G45−pz4G35)− δij

2 (G35+G45),

Au(i, j) = 2¡pi4pj₃G14G23+pi3pj₃G23G45+pi4pj₄G14G35¢+

+_√δij

s(pz3G45−pz4G35)− δij

2 (G35+G45). (3.48)

The propagator factors are given by:

G13 = ¡ 1

(p1+p3)2−m23

¢, _G23= ¡ 1

(p2+p3)2−m23

¢,

G14 = ¡ 1

(p1+p4)2−m24

¢, _G24= ¡ 1

(p2+p4)2−m24

¢,

G35 = _¡ 1

(p3+p5)2−m24+im4Γ4¢

, _G45= _¡ 1

(p4+p5)2−m23+im3Γ3¢

JHEP04(2002)028

For process (h), the matrix element is given by:

|M|2_(e−

1e+2 →h03A04, H30A04) =e4sp2T ·

L2 ν

¡

L2 e+R2e

¢ ¡

cos2_{(β−α),sin}2_(β−α)¢

(s₋M2

Z)2+MZ2Γ 2 Z

. (3.50)

We have used the same notation as in eqs. (3.9). The coupling factor cos2(β ₋α) is for

h0A0 production whereas sin2(β₋α) is forH0A0 production.

3.6 Charged Higgs production

For hadronic collisions, the following production modes are implemented:

gb _−→ tH−+ c.c. ; (3.51)

gg _−→ t¯bH−+ c.c. ; (3.52)

qq¯_−→ t¯bH−+ c.c. ; (3.53)

gg _−→ eteb∗_H−_{+ c.c. ; (3.54)}

qq¯_−→ eteb∗H−+ c.c. ; (3.55)

qq¯_−→ H+H−; (3.56)

b¯b _−→ W±H∓; (3.57)

qb _−→ q0bH++ c.c.. (3.58)

Charged Higgs bosons are also produced through the decay of heavier objects including gauginos and third generation sfermions. Perhaps most notably, an important potential production mode for charged Higgs bosons is through the decay of top quarks and, as with other decay processes, this is achieved by specifying this decay mode in the input file. The decay is carried out using the three-body matrix element. This matrix element is not included in the spin correlation algorithm [17] which is the default treatment from

version 6.4. If this decay is required, the matrix element code NME=200 should be used

and spin correlations and supersymmetric three body decays should be switched off, i.e.

SYSPIN=.FALSE.andTHREEB=.FALSE..

When simulating processes (3.52), (3.53) and (3.58) one should bear in mind that there is potentially a large contribution from top quark decay when this decay mode is available. In this case the user should choose to simulate either the top quark production or the relevant three-body process in order to avoid double counting. In the limit in which the

bottom quark is generated by the collinear splitting g _→ b¯b, process (3.52) is described

by the two body process (3.51). This contribution becomes significant if the mass of the charged Higgs boson is comparable or greater than the top quark mass. Again, the user should choose to simulate either of the two processes in order to avoid double counting.

For leptonic collisions, in addition to the top quark decay, charged Higgs bosons can be produced in a process analogous to process (3.56):

e−e+_−→H+H−. (3.59)

The matrix element is obtained immediately from the hadronic case by changing the

ap-propriate photon andZ0 _{couplings and removing the colour factor 1/N}

JHEP04(2002)028

Process (3.51) is given by:

|M|2_(g

1b2 →t3H4−) =

µ

g2 s

2NC

¶ µ

e2

2 sinθ2

W

¶

|VCKM[tb]|2

µ

m2_btan2β+m2_t cot2β

2M2

W

¶

×

×

µ

−u2₄ st3

¶ ·

1 + 2m

2 4−m23

u4

µ

1 + m

2 3 t3 +m 2 4 u4 ¶¸ . (3.60)

Process (3.52) is evaluated using helicity amplitudes in an optimized basis, similarly

to the evaluation of the neutral Higgs boson production processgg _→QQ¯Φ in section 3.5.

Process (3.53) is evaluated using a more general formula for q1q¯2 → Q3Q¯4Φ which is

given by:

|M|2_(q

1q¯2→Q3Q¯4Φ) =

=

µ

g4 sCF

2NC

¶

2φ2

H

s

" ¡

p2_T3 +p2_T4−p2_T5¢+ 4p0·p3 p0·p4|G|2 +

+ 2 (p3·p4−λm3m4)£−s|G|2+ 2 Re¡GG∗3p2T4 +GG

∗

4p2T3− G3G

∗ 4p2T5

¢¤

−

− 4 Re

"

GG3∗p2T4+GG

∗

4p2T3+G(G

∗

3p0·p4+G4∗p0·p3)

p2_T3+p2_{T4 −}p2_T5

2

#

−

−4 Im [_GG3∗pz4 − GG∗4pz3]λ0

√

s(px4py3 −px3py4)

#

, (3.61)

with p0 = p1 +p2. Denoting the Q3Q¯4Φ vertex as (gS+gPγ5), the various coupling

coefficients appearing above are defined by:

φ2_H =g_S2+g2_P, λ = g

2 S−g2P

φ2 H

,

λ0 = 2gSgP

φ2_H . (3.62)

Process (3.54) is evaluated using helicity amplitudes. The expressions are the same as eqs. (3.47)–(3.49) but the coupling coefficient needs to be changed. Let us consider the

vertex involving an incoming H+ _{and outgoing et}

ieb∗j. We have the following replacement:

g_QeM2_e

Q

M2 W

−→ √1

2

·µ

sin 2β₋m

2

btanβ+m2t cotβ

M2 W

¶

Qt_LiQb_Lj−

− mtmb

M2 W

(tanβ+ cotβ)Qt_RiQb_Rj− mb

M2 W

(₋µ+Abtanβ)QtLiQbRj−

−_Mm2t W

(₋µ+Atcotβ)QtRiQbLj

¸

. (3.63)

Process (3.56) is given by:

|M|2 ₌ e4sp2T

NC ×

×   ¯ ¯ ¯ ¯ ¯

QqQH+

s +

LqLH+

s₋M2

Z+iMZΓZ

¯ ¯ ¯ ¯ ¯ 2 + ¯ ¯ ¯ ¯ ¯

QqQH+

s +

RqLH+

s₋M2

Z+iMZΓZ

¯ ¯ ¯ ¯ ¯ 2

JHEP04(2002)028

We have used the same notation as in eq. (3.9). QH+ = +1, and L_H+ is given by eq. (3.6)

withI_f3= +1/2. For the leptonic case the subscriptsq should be replaced withe, and the

colour factor 1/NC should be replaced by 1.

The matrix element for process (3.57) is as given in [18].

Process (3.58) is given in [19]. We use a more compact expression which sets the

kinematic bottom quark mass to zero. For the processb1U2 →b3D4H5+ we have:

|M|2_(b

1U2 →b3D4H5+) =

1 2

µ

e2

2xW

¶3

|VCKM[U D]|2

1

(M2

[24]−MW2 )2 ×

×h|VCKM[tb]|2|At|2+|AΦ|2+ ReVCKM[tb]·2A∗_ΦAt)

i

;

|At|2 = 2M[12]|Gt|2 2

 µmbtβ

mW

¶2

[3545]

2 +

Ã

mtt−_β1

mW

!2

m2_tM_[34]2

 ;

|AΦ|2 = (Gh,H2 +GA2)

³

−M_[13]2 ´[2545]

2 ;

Re(2A∗

ΦAt) =

mbtβ

MW

(_Gh,H +_GA)_×

×h2 Re (_Gt)³M_[45]2 ₋M_H2±

´ ³

M_[12]2 ´ ³₋M_[13]2 ´₋

−Re(_Gt)M_[45]2 [1243] + Im(_Gt)M_[45]2 _·4 det(1234)i. (3.65)

Here we have defined [abcd] = 4(pa·pb pc·pd+pa·pd pb·pc−pa·pcpb·pd) and det(abcd) =

det(pµa, pνb, p

ρ c, pσd) =

P

µνρσ²µνρσpµapνbp ρ

cpσd where²0123 = +1.

The propagators are defined by:

Gt = 1

M2

[35]−m2t +imtΓt

= M

2

[35]−m2t −imtΓt

(M2

[35]−m2t)2+m2tΓ 2 t

;

Gh,H = mbsinαcos(β−α)/MW cosβ

M2

[13]−Mh2

+mbcosαsin(β−α)/MWcosβ

M2

[13]−MH2

;

GA = mbtanβ/MW

M2

[13]−MA2

.

Processes with antiquarks in the initial state can be obtained by exchanging the subscripts

(1 _↔ 3) for the subprocess ¯bU _→ ¯bDH+_{, (2 ↔ 4) for the subprocess bD}¯ _{→ bU H}¯ +_{, and}

both for the subprocess ¯bD¯ _→ ¯bU H¯ +. Note that the definitions of the invariant masses

squared are such that these imply momentum substitutions of the form (p1↔ −p3).

3.7 R-parity violating processes

The cross sections for the implemented R-parity violating hadron-hadron processes are

presented in ref. [20]. In hadron-hadron collisions only those 2 _→ 2 processes that can

occur via a resonants-channel sparticle exchange are included, i.e. processes that can only

occur via a t-channel diagram or for which the s-channel resonance is not kinematically

JHEP04(2002)028

Ine+e− collisions we include all the possible 2_→2 single sparticle production

mecha-nisms, even if there is nos-channel resonance, and a limited set of 2_→2 processes where

the final state contains two SM particles. As with the hadron-hadron processes, there

are four types of process: single gaugino production; single slepton production in

associa-tion with a gauge boson; single slepton producassocia-tion in associaassocia-tion with a Higgs boson; SM

particle production.

3.7.1 Single gaugino production

In e+e− _{collisions, there are two possible production mechanisms for a gaugino and a}

lepton, eithere+e− _→χe0_{ν ore}+_e−_→χe+_`−_{. The matrix elements for these processes have}

been calculated previously in refs. [21, 22, 23]. We have recalculated these matrix elements and the results are in agreement with [23]. We also agree with refs. [21, 22], which do not include sfermion mixing, if we set our mixing to zero. All matrix elements are averaged over the spins of the incoming leptons as before.

|M|2_(e−

i e+i →νjχe0l) =

= λ

2 iji

2

"

|a(ν_e∗

j)|2sR(νej)

³

s₋M2

e

χ0 l

´

+ 1

(u₋M2

`iL)

2

h

|a(e`∗

i1)|2+|b(`e∗i1)|2

i

u³u₋M2

e

χ0 l

´

+

+ 1

(t₋M2

`iR)

2

h

|a(`ei2)|2+|b(`ei2)|2

i

t³t₋M_χe20 l

´

+

+ 2

(u₋M2

`iL)

a(_eν_j∗)a(`e∗_i1)R(ν_ej)

³

s₋m2_eνj´su+

+ 2

(u₋M2

`iL)(t−M

2 `iR)

a(`e∗<sub>i1</sub>)a(`ei2)ut+

+ 2

(t₋M2

`iR)

a(_eν_j∗)a(`ei2)R(eνj)

³

s₋m2_νej´st

#

,

|M|2_(e−

i e+i →`+jχe−l ) =

= g

2_λ2 iji

4



R(_eνj)sh¡|a(eνj∗)|2+|b(eνj∗)|2

¢ ³

s₋m2_{`j −}M2

e

χ−_l

´

−4a(_eν_j∗)b(ν_ej∗)m`jMχ_e−_l

i

+

+_³ 1

u₋M2

e

νi

´2

³

M2

e

χ−_l −u

´ ³

m2_{`j −}u´ £_|a(_eν_i∗)_|2+_|b(ν_ei∗)_|2¤₋

−³ 2s

u₋M2

e

νi

´³s₋M_νe2_j´R(ν_ej)a(νei∗)

h

a(ν_ej∗)u+m`jMχ_e−_l b(νej∗)

i

. (3.66)

The couplingsa, b of the sleptons to the gauginos are given in ref. [20], and

R(˜a) = _¡ 1

JHEP04(2002)028

There are a number of production processes for a slepton in association with a gauge boson. We have confirmed the cross section for these processes, which were first calculated in ref. [24]. The sneutrino resonance is not usually accessible for these processes as the charged

slepton-sneutrino mass difference is less than the W± _{mass. However, the resonance can}

be accessible for the third generation due to left-right mixing of the staus.

The matrix elements for these processes are given below, including left-right sfermion mixing:

|M|2_(e−

i e+i →γνej) =

e2λ2_iji

2

·

u t +

t u +

2

ut

³

M_νe2_{j −}u´ ³M_eν2_{j −}t´¸,

|M|2_(e−

i e+i →W+`ejα) =

= g

2_λ2_iji|Lj 1α|2

2M2

W

·

s2p2_cmR(_eνj) +

1

4u2

h

2M_W2 ³ut₋M_W2 M_e2

`jα

´

+u2si₋

−₂s_uR(_eνj)

³

s₋M_eν2_j´ hM_W2 ³2M_e2

`jα−u

´

+u³s₋M_e2

`jα

´ii

,

|M|2_(e−

i e+i →Z0νej) =

= g

2_λ2 iji

cos2_θ

WM_Z2

"

|Z_ν11_j|2s2p2_cmR(_eνj) +

Z_e2_R t2

h

2M_Z2³ut₋M_νe2_jM_Z2´+st2i+

+ Z

2 eL

u2

h

2M_Z2³ut₋M_νe2_jM_Z2´+su2i+

+ 2ZeLZeR

ut

h

2M_Z2³M_νe2

j −t

´ ³

M_νe2

j −u

´

−suti₋ (3.68)

−Z 11

e

νjZeR

t sR(νej)

³

s₋M_eν2_j´ hM_Z2³2M_νe2_{j −}t´+t³s₋M_νe2_j´i+

+Z

11

e

νjZeL

u sR(νej)

³

s₋M_eν2_j´ hM_Z2³2M_νe2_{j −}u´+u³s₋M_eν2_j´i

#

,

wherepcm is the momentum of the final-state particles in the centre-of-mass frame. As we

neglect the electron mass, the cross section for e−_i e+_{i →} γν_ej is divergent as u, t → 0, i.e.

in the limit that the photon is collinear with the incoming lepton beams, and therefore a cut on the transverse momenta of the outgoing particles is used to regularize the

diver-gence for this process. The couplings of the Z0 _{to the leptons and sleptons are given in}

ref. [20].

3.7.3 Slepton production in association with a Higgs boson